Journal of Symplectic Geometry

Volume 6 (2008)

Number 1

A groupoid approach to quantization

Pages: 61 – 125

DOI: http://dx.doi.org/10.4310/JSG.2008.v6.n1.a4

Author

Eli Hawkins

Abstract

Many interesting $C∗$-algebras can be viewed as quantizations of Poisson manifolds. I propose that a Poisson manifold may be quantized by a twisted polarized convolution $C∗$-algebra of a symplectic groupoid. Toward this end, I define polarizations for Lie groupoids and sketch the construction of this algebra. A large number of examples show that this idea unifies previous geometric constructions, including geometric quantization of symplectic manifolds and the $C∗$-algebra of a Lie groupoid. I sketch a few new examples, including twisted groupoid $C∗$-algebras as quantizations of bundle affine Poisson structures.